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We once got to Fife via train and bike from London - but, if you are not feeling up for cycling against the sea wind, there are other ways to get there...
From Edinburgh (which you can reach by train or plane) there are several ways to get to Fife.
You can also easily reach Fife from Glasgow via the same methods.
Car
If you plan to visit the area, the best way to get around is by car.

You can rent one from Edinburgh airport and it will be about a 1h drive depending on where you are staying.
From Glasgow it will also be +/- 1 hour drive time

Taxi
You can also take a taxi from Edinburgh Airport to the place you are staying via Edinburgh Airport Taxi. You can get a quote online, but depending on the vehicle chosen and final destination it will be around £100+.
Taxis from Glasgow are also possible and equally priced.
Train/Buses
From both the centre of Edinburgh and the airport, you can take the train to different towns in Fife: Leven, Cupar and Leuchars. From there you can book a taxi or use local buses.
Direct buses run from Leven to coastal towns, including Anstruther. It is around a 1 hour bus ride.
From Cupar and Leuchars, you will need to take two buses to get to coastal towns, both going via St Andrews.
Train journeys from Glasgow are similar and also go via Edinburgh.